The Stokes number, St, used in particle-dynamics studies, is a dimensionless combination of five variables: acceleration of gravity g, viscosity μ, density p, particle velocity U, and particle diameter D. (a) If St is proportional to μ and inversely proportional to g, find its form. (b) Show that St is actually the quotient of two more traditional dimensionless groups.
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- The Stokes number, St, used in particle dynamics studies,is a dimensionless combination of five variables: accelerationof gravity g , viscosity μ , density ρ , particle velocity U ,and particle diameter D . ( a ) If St is proportional to μand inversely proportional to g , find its form . ( b ) Showthat St is actually the quotient of two more traditionaldimensionless groups.The Stokes number, St, used in particle-dynamics studies is a dimensionless combination of 5 variables: acceleration of gravity g, viscosity μ, density p, particle velocity U, and particle diameter D. If St is propotional to μ and inversely proportional to g, find its dimensionless form.The spin rate of a tennis ball determines the aerodynamic forces acting on it. In turn, the spin rate is a§ectedby the aerodynamic torque. If the torque depends on áight speed V , density , viscosity , ball diameter D,angular velocity !, and the fuzz height, hf , Önd the important dimensionless variables for this case. Use V ,, and D as your scaling (repeating) variables.
- When a steady uniform stream flows over a circular cylinder, vortices are shed at a periodic rate. These are referred to as Kármán vortices. The frequency of vortex shedding få is defined by the free-stream speed V, fluid density p, fluid viscosity u, and cylinder diameter D. Use the Buckingham Pi method to show a dimensionless relationship for Kármán vortex shedding frequency is St = f (Re). Show all your work. V DDuring World War II, Sir Geoffrey Taylor, a British fluid dynamicist, used dimensional analysis to estimate theenergy released by an atomic bomb explosion. He assumed that the energy released E, was a function of blastwave radius R, air density ρ, and time t. Arrange these variables into single dimensionless group, which we mayterm the blast wave number.When a liquid in a beaker is stired, whirlpool will form and there will be an elevation difference h, between the center of the liquid surface and the rim of the liquid surface. Apply the method of repeating variables to generate a dimensional relationship for elevation difference (h), angular velocity (@) of the whirlpool, fluid density (p). gravitational acceleration (2), and radius (R) of the container. Take o. pand R as the repeating variables.
- The velocity V of propagation of ripples on the surface of a shallow liquid depends on the gravitational acceleration g and the liquid depth h. If Buckingham's Theorem is used to identify the salient dimensionless group(s), how many dimensionless group(s) will be obtained? Number of dimensionless group(s) = 1. {1} (Enter your answer as a number.)Evaluate the use of dimensionless analysis using the Buckingham Pi Theorem for a given fluid flow system (D4) , where resistance tomotion ‘R’ for a sphere of diameter ‘D’ moving at constant velocity on the surface of a liquid is due to the density ‘ρ’ and the surfacewaves produced by the acceleration of gravity ‘g’. The dimensionless quantity linking these quantities is Ne= Function (Fr). To do thisyou must apply dimensional analysis to fluid flow system given in Figure 1 (P11). PICTURE IS ALSO ATTACHEDThe wall shear stress Twin a boundary layer is assumed to be a function of stream velocity U, boundary layer thickness , local turbulence velocity u', density p, and local pressure gradient dp/dx. Using (p, U, and ) as repeating variables, rewrite this relationship as a dimensionless function.
- The power P generated by a certain windmill design depends upon its diameter D, the air density p, the wind velocity V, the rotation rate 0, and the number of blades n. (a) Write this relationship in dimensionless form. A model windmill, of diameter 50 cm, develops 2.7 kW at sea level when V= 40 m/s and when rotating at 4800 r/min. (b) What power will be developed by a geometrically and dynamically similar prototype, of diameter 5 m, in winds of 12 m/s at 2000 m standard altitude? (c) What is the appropriate rotation rate of the prototype?1. The thrust of a marine propeller Fr depends on water density p, propeller diameter D, speed of advance through the water V, acceleration due to gravity g, the angular speed of the propeller w, the water pressure p, and the water viscosity μ. You want to find a set of dimensionless variables on which the thrust coefficient depends. In other words CT = FT · = ƒen(#1, #2, ...) pV2D2 (a) What is k? Explain. (b) Find the 's on the right-hand-side of equation 1 if one of them HAS to be a Froude number gD/V², (1)I want handwritten don't copy from chegg same to same change it even if u copy and handwritten only